the instantaneous power demand from the loads therein and branch impedances. Hence, this technique requires that every node is a SN.
Distributed Optimal Reactive Power Flow Control (DORPF) [7, 28] requires that DESs are grouped into possibly overlapping clusters and that, for each of them, one of the nodes becomes the cluster head (CH). Also, within each cluster, the gradient of the power distribution loss is estimated through local measurements. Relying on the estimated gradient, the CH computes the set of reactive powers that have to be injected by the two DESs in its own cluster (one being associated with the CH) in order to minimize the distribution power losses and spreads this information among its neighboring DESs. While different clustering procedures are possible, as stated in [7], the most effective clustering technique is the one proposed in [6] (see CBSC above). This technique requires that only the nodes that are connected to DESs are SNs, thus relaxing the requirements on the nodes in terms of communication and complexity. Due to this, the same clustering approach of CBSC is also considered for DORPF.
3.3
Token Ring Control
CBSC and DORPF require that groups of nodes iteratively take a control action (i.e., inject a certain amount of power in the grid) in order to reduce as much as possible the distribution power loss. The PCC is considered as a SN during the optimization process and its identifier is 0. The procedure of having at any given time a single SN allowed to modify the injected current, before letting the next SN to operate is similar to the token ring approach widely used in communication networks. For the sake of clarity, we recall that the access to the communication medium is arbitrated through a special packet called token. At any given step, one of the SNs owns the token, being in charge of implementing the control action and communicating with other SNs. All the other SNs are only allowed to answer explicit requests from the token owner, but are not allowed to contact it in the first place. For what has been said so far, the communication network can be considered collision free. Therefore, we will use here the related terminology, where however token ownership is associated to the current control, rather than to the possibility to transmit information. Indeed, when a node has the token it may communicate (in a two-way fashion) with other nodes in order to collect the information need for the control action. However, only SN with the token initiates the
communications, while other nodes are only allowed to answer its requests. When the current tokenowner releases the token, the next owner is chosen according to a specific policy. Note that choosing the next owner corresponds to establishing an order (also referred to as control scheduling) for the execution of the control actions. Two policies for the owner selection are now discussed. The first ordering strategy aims at maximizing the convergence rate of the optimization algorithms. The second ordering strategy is considered as a comparison reference and simply aims at minimizing the length of the token path in each token round, i.e., minimize the communication overhead needed to move the token.
Heuristic for Convergence Rate Maximization: improving the convergence rate of the considered optimization algorithms has two main benefits. First, the optimization becomes more responsive to changes in the power demand from the loads. Second, further power is saved during optimization. The convergence rate can be improved by suitably tuning the order in which nodes perform the control action, i.e., defining a new token owner selection rule. The optimal (in the sense of maximum convergence rate) selection rule requires that at least one SN has a full knowledge of the network state, but, in this case, a centralized optimization approach would be the best choice. For this reason a heuristic selection rule, that does not increase the amount of information that each SN has to collect for the optimization purpose, is proposed. This rule is based on the observation that updates in clusters with a higher power demand should have a larger impact on the total power loss.
In details, any two clusters are referred to as adjacent if at least one pair of nodes belonging to the two clusters is connected by a line with no nodes in between. Hence, if a node belonging to the two clusters exists, the two clusters are adjacent. At the beginning of the optimization process, the token owner is uniformly chosen at random among the SNs by the PCC. At each optimization step, the token owner collects information about the actual power demand of all the adjacent clusters. The token is then passed to the head of the cluster with the highest power demand. If more CHs are eligible, one of them is chosen uniformly at random.
Token Path Length Minimization: as a baseline strategy we consider that obtained by minimizing the length of the token path. To this end, suitable SNs identifiers and a next owner updating rule have to be defined. The identifiers have to be assigned starting from 1 and visiting the nodes with a depth first pre-ordered tree traversal. If the current token owner is the SN with identifier i, then the next owner identifier will be j = (i + 1) mod N .
3.4. Electrical Grid Topology Generation 45